Critical Curves and Caustics of Triple-lens Models Kamil Danˇek and David Heyrovsky´ Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic; 5 [email protected] 1 0 2 [email protected] n u J ABSTRACT 5 1 Among the 25 planetary systems detected up to now by gravitational mi- ] P crolensing, there are two cases of a star with two planets, and two cases of a E . binary star with a planet. Other, yet undetected types of triple lenses include h p triple stars or stars with a planet with a moon. The analysis and interpretation - o of such events is hindered by the lack of understanding of essential characteristics r t of triple lenses, such as their critical curves and caustics. We present here analyt- s a ical and numerical methods for mapping the critical-curve topology and caustic [ cusp number in the parameter space of n-point-mass lenses. We apply the meth- 2 ods to the analysis of four symmetric triple-lens models, and obtain altogether 9 v 9 different critical-curve topologies and 32 caustic structures. While these results 1 5 include various generic types, they represent just a subset of all possible triple- 6 lens critical curves and caustics. Using the analyzed models, we demonstrate 0 . interesting features of triple lenses that do not occur in two-point-mass lenses. 1 0 We show an example of a lens that cannot be described by the Chang-Refsdal 5 1 model in the wide limit. In the close limit we demonstrate unusual structures : v of primary and secondary caustic loops, and explain the conditions for their oc- i X currence. In the planetary limit we find that the presence of a planet may lead r to a whole sequence of additional caustic metamorphoses. We show that a pair a of planets may change the structure of the primary caustic even when placed far from their resonant position at the Einstein radius. Subject headings: gravitational lensing: micro — methods: analytical — plane- tary systems – 2 – 1. Introduction In the past two decades gravitational microlensing surveys have been very successful, in particular as a tool for studying the stellar population toward the Galactic bulge. In a microlensing event, a star passing close to the line of sight to a background “source” star is detected by its gravitational lens effect, which temporarily amplifies the flux from the source (e.g., Paczyn´ski 1996). The main advantage of the method is its sensitivity to low-mass objects, ranging from stellar down to planetary masses, with most of them too faint to be routinely detected by other means. In addition to single-star microlensing events, right from their start of operations the surveys have detected events with binary-star lenses (Udalski et al. 1994; Alard et al. 1995). Thefrequencyofbinaryeventsislowerthanthefrequencyofbinarystars, sincebinarieswith too close or too far components often mimic single-lens events. The microlensing sensitivity to low mass ratios finally led in 2003 to the first detection of microlensing by a star with a planet (Bond et al. 2004). By the time of writing, altogether 25 planetary systems had been detected by microlensing1. A majority involved a star with a single planet; nevertheless, four of them involved three-body systems. In two cases the lens was a star with two planets (Gaudi et al. 2008; Han et al. 2013), while the other two involved a binary with a planet (Gould et al. 2014; Poleski et al. 2014). Other possible triple-lens systems that had not been detected yet include triple stars, or even lenses formed by a star with a planet with a moon. Turning to the underlying physics, in the case of a single lens the light from the source star is split into two images, which remain generally unresolved due to their small angular separation (cid:46) 1 mas. The accompanying temporary amplification of the flux from the source typically produces a simple symmetrically peaked light curve (e.g., Paczyn´ski 1996). Lenses with multiple components produce a higher number of images and lead to a greater diversity of light curves, which peak anytime the source crosses or approaches the caustic of the lens. Any simple caustic crossing leads to the appearance or disappearance of a pair of unresolved images at positions defining the critical curve of the lens. The caustic and the critical curve thus are key characteristics of the lens. They depend sensitively on the lens parameters: the masses and positions of the components. Understanding the range of possible critical-curve and caustic geometries is a prerequisite for successful analysis and interpretation of observed microlensing light curves. Microlensing by a two-component lens, such as a binary star or a star with a planet, is well described by the two-point-mass lens model. Such a lens has only two relevant 1http://exoplanetarchive.ipac.caltech.edu – 3 – parameters: the mass ratio, and the projected component separation. The model has been analyzed in detail by Schneider & Weiss (1986), Erdl & Schneider (1993), and Witt & Petters (1993); its limiting cases were studied by Dominik (1999). For any mass ratio the critical curve was shown to have three different topologies. In order of decreasing component separation these are: “wide” with two separate loops, “intermediate” (or resonant) with a single loop, and “close” with an outer plus two inner loops. Each regime has a corresponding caustic geometry with the same number of separate non-overlapping loops: wide has two four-cusped loops, intermediate a single six-cusped loop, and close has one four-cusped plus two three-cusped loops. A source positioned outside the caustic has three images, while a source positioned inside the caustic has five images. Microlensing by triple lenses can be described analogously by the three-point-mass lens model. There are five lens parameters: two relative masses, and three relative positions defining the two-dimensional configuration of the components in the plane of the sky. In comparison with the two-point-mass lens, the model has a number of qualitative differences. For example, varying the lens parameters may lead to a change in the cusp number of the caustic without any accompanying change in the topology of the critical curve. Such changes in the caustic structure occur via swallow-tail or butterfly metamorphoses (Schneider et al. 1992). In addition, loops of the caustic may overlap, and individual loops may self-intersect. As a result, the caustic may have inner multiply nested regions. All caustics separate the outer four-image region from an inner six-image region. Only caustics with overlapping loops or self-intersections have additional eight-image regions, in case of double nesting even ten-image regions. However, unlike in the two-point-mass lens case, the full range of critical-curve topolo- gies and caustic structures of the three-point-mass lens has not been explored yet. Such a studywouldrequireasystematicmappingofthefive-dimensionalparameterspace, detecting changes in the critical curve and caustic of the corresponding lens. Nevertheless, a range of published works have explored different specific regimes of triple lenses. The first studies explored binary lenses with an additional external shear (Grieger et al. 1989; Witt & Petters 1993), with caustics already displaying swallow tails and butterflies. Most numerous are the studies involving a stellar lens with two (or more) planets. Some of them demonstrate effects on the light curve, others for example place constraints on the presence of a second planet in observed single-planet events. Without claiming completeness, we refer here to the works of Gaudi et al. (1998), Bozza (1999), Han et al. (2001), Han & Park (2002), Rattenbury et al. (2002), Han (2005), Kubas et al. (2008), Asada (2009), Ryu et al. (2011), Song et al. (2014), and Zhu et al. (2014). Lensing by a binary star with a planet has been explored less frequently (Bennett et – 4 – al. 1999; Lee et al. 2008; Han 2008a; Chung & Park 2010). In view of the two recently detected systems, this is bound to change. The close and far limits of triple-star lensing were investigated by Bozza (2000a,b). Finally, lensing by a star with a planet with a moon was studied by Han & Han (2002), Gaudi et al. (2003), Han (2008b), and Liebig & Wambsganss (2010), with prospects for detecting such systems remaining open. Here we set out to systematically study the critical curves and caustics of triple lenses. BasedontheworkofErdl&Schneider(1993)andWitt&Petters(1993),wedevelopmethods for efficient mapping of critical-curve topologies and caustic geometries in the parameter space of n-point-mass lenses. We then apply the methods to the analysis of simple triple- lens models. Initial steps of this research appeared in Danˇek (2010), Danˇek & Heyrovsky´ (2011), and Danˇek & Heyrovsky´ (2014). We start in § 2 by introducing the basic concepts of n-point-mass lensing. In particular, we concentrate on the Jacobian and its properties, such as the equivalence of its contours with critical curves of re-scaled lens configurations (Danˇek & Heyrovsky´ 2015). Analytical and numerical methods for mapping critical-curve topologies in the lens parameter space are introduced in § 3. In § 4 we discuss the caustic and its metamorphoses, and show how to track changes in cusp number using the cusp and morph curves (Danˇek & Heyrovsky´ 2015). We apply the methods to triple lenses in § 5, starting with a brief overview of their properties in § 5.1. In § 5.2 – § 5.5 we include a full analysis of four symmetric two-parameter triple-lens models, with an overview of the found critical curves and caustics in § 5.6. We end by summarizing the main results and highlights in § 6. 2. The n-point-mass lens and its Jacobian Galactic gravitational microlensing events can be described using a simple n-point-mass lens model, consisting of n components (stars, planets) in a single lens plane with no external shear and no convergence due to continuous matter. Following Witt (1990) we describe positions in the plane of the sky as points in the complex plane, with separations measured in units of the Einstein radius corresponding to the total mass of the lens. The relation between the position of a background source ζ and the position z of its image formed by the lens is expressed by the lens equation n (cid:88) µ j ζ = z − , (1) z¯−z¯ j j=1 where z and µ are the positions and fractional masses, respectively, of the individual j j components of the lens, and bars over variables denote their complex conjugation. The – 5 – complex plane of the image positions z is called the image plane, and the complex plane of the source positions ζ is called the source plane. The fractional masses are normalized to the total mass of the lens, so that (cid:80)n µ = 1. j=1 j The positions of individual components z generally change with time as they orbit j around the center of mass of the lens. In this work we study the lensing properties of a “snapshot” n-body configuration at a given instant. The case when the change of lens parameters is non-negligible on the timescale of the microlensing event can be described by a corresponding sequence of such snapshot configurations. The n-point-mass lens with n > 1 components produces between n + 1 and 5(n − 1) images (in steps of two) of any point in the source plane (Rhie 2003; Khavinson & Neumann 2006). In the image plane images appear and disappear in pairs along the critical curve of the lens. The critical curve can be expressed as the set of all points z for which the sum cc n (cid:88) µ j = e−2iφ, (2) (z −z )2 cc j j=1 lies on the unit circle (Witt 1990). Here φ is a phase parameter spanning the interval φ ∈ [0,π). In the source plane the image number changes when the source crosses the caustic ζ of the lens. The caustic is obtained by tracing critical-curve points back to the c source plane using equation (1), i.e., n (cid:88) µ j ζ = z − . (3) c cc z¯ −z¯ cc j j=1 In mathematical terms, the critical curve is the set of points in the image plane with zero Jacobian of the lens equation. The Jacobian (cid:12)(cid:12)(cid:88)n µ (cid:12)(cid:12)2 detJ (z) = 1−(cid:12) j (cid:12) (4) (cid:12) (z −z )2 (cid:12) (cid:12) j (cid:12) j=1 is discussed in detail in Danˇek & Heyrovsky´ (2015). Here we only summarize its properties important for the following analyses. As seen from equation (4), the Jacobian is a real functiondefinedoverthecomplexplane, runningfrom−∞atthepositionsofallcomponents z = z to 1 at complex infinity and at the positions of all Jacobian maxima. These can be j found as the roots of the polynomial obtained from n (cid:88) µ j = 0. (5) (z −z )2 j j=1 – 6 – The degree of the corresponding polynomial indicates the Jacobian has up to 2n−2 different maxima. The number may be lower if there are any degenerate roots; these correspond to higher-order maxima. A doubly degenerate root corresponds to a double maximum, a root with degeneracy 3 corresponds to a triple maximum, etc. The saddle points of the Jacobian can be found similarly among the roots of the poly- nomial obtained from n (cid:88) µ j = 0. (6) (z −z )3 j j=1 The corresponding polynomial has up to 3n−3 different roots. First we need to sort out any common roots of equations (5) and (6), those correspond to higher-order maxima instead of saddles. All remaining roots are Jacobian saddle points. The number of different saddles may be reduced further if there are any degenerate roots; these identify higher-order saddles. A non-degenerate root corresponds to a simple saddle, a doubly degenerate root corresponds to a monkey saddle, etc. Studying the contours of the Jacobian, Danˇek & Heyrovsky´ (2015) pointed out a re- markable correspondence. While the zero-Jacobian contour is the critical curve z of the cc lens, any other Jacobian contour z with detJ (z ) = λ is a re-scaled critical curve of a lens λ λ with the same components in re-scaled positions. Denoting by z (µ ,z ) and z (µ ,z ) the cc j j λ j j critical curve and detJ = λ contour, respectively, of a lens with masses µ and positions z , j j we can express this correspondence by √ √ z (µ ,z ) = z (µ ,z 4 1−λ)/ 4 1−λ. (7) λ j j cc j j For λ = 0 we get the critical curve of the original configuration. Close to the original positions, the λ → −∞ contours are shrunk versions of the wide-limit critical curves. At the other limit, the highest-Jacobian λ → 1 contours are expanded versions of the close-limit critical curves. A single Jacobian contour plot thus yields the critical curves for all scalings of the given lens configuration, from the close to the wide limit. 3. Critical-curve topology regions in parameter space and their boundaries 3.1. Critical-curve topology and its changes Here we summarize general properties of n-point-mass lens critical curves following Danˇek&Heyrovsky´ (2015). Varyingthephaseparameterφinthecriticalcurveequation(2), we obtain 2n continuous line segments, which may connect to form 1 ≤ N ≤ 2n closed loops – 7 – loops2 of the critical curve (Witt 1990). Individual loops may lie in separate regions of the image plane, or they may lie nested inside other loops. The total number and mutual position of loops define the topology of the critical curve. In the wide limit, all lenses have a critical curve with N = n separate loops, corre- loops sponding to n Einstein rings of the individual components. In the close limit, the critical curve has N = 1 + N loops, where N is the number of different Jacobian max- loops max max ima. One loop corresponds to the Einstein ring of the total mass, plus there is a small loop around each maximum of the Jacobian. In view of the discussion following equation (5), N ≤ 2n−2, so that in the close limit N ≤ 2n−1. The equality holds if the Jacobian max loops has only non-degenerate (simple) maxima. Any degenerate (higher-order) maximum reduces the number of loops. For example, the two-point-mass lens always has two different maxima, thus its critical curve always has three loops in the close limit. When varying the parameters of the lens such as its scale, the topology may undergo changeswhenindividualloopsmergeorsplit. ThisoccursatJacobiansaddlepointswhenthe criticalcurvepassesthroughthem(e.g.,Erdl&Schneider1993). Twoloopscomeintocontact at a simple saddle, three at a monkey saddle, more loops at gradually higher-order saddles. As shown by equation (6), the Jacobian may have up to 3n − 3 different saddle points, with the highest number occurring when there are no higher-order saddles and no higher- order maxima. The number of different Jacobian contours passing through the set of saddles identifies the total number of changes in critical-curve topology encountered when varying the scale of the lens from the wide to the close limit. This can be seen as a consequence of the Jacobian-contour / critical-curve correspondence expressed by equation (7). Therefore, the critical curve of an n-point-mass lens may undergo no more than 3n − 3 changes in topology between the wide and close limits. In the case of the two-point-mass lens, there are always three simple saddles. One lies on the axis between the components, while an off-axis pair of saddles lies on a different Jacobian contour. The two distinct saddle contours imply that the two-point-mass lens critical curve always undergoes two changes when varying the component separation s, and thushasexactlythreetopologies. Thewidetopologyhastwoseparateloops,theintermediate topology has a single merged loop, and the close topology has an outer loop plus two small inner loops around the Jacobian maxima. The topology sequence is independent of the second lens parameter, the mass ratio of the lens components. Proceeding to lenses with more than two components, we note that the shape of the 2We note that at least for n < 3 the sharp upper bound is 2n−1, and even for triple lenses we have found no more than 5 loops so far. – 8 – critical curve depends on 3n−4 lens parameters (e.g., Danˇek & Heyrovsky´ 2015). Follow- ing the preceding discussion, boundaries between regions in parameter space with different critical-curve topology can be found by identifying parameter combinations, for which the critical curve passes through a saddle point of the Jacobian (e.g., Erdl & Schneider 1993). The search for these boundaries is thus mathematically reduced to finding the conditions for the occurrence of a common solution of equations (2) and (6) . The usual analytical approach described in the following § 3.2 is based on rewriting both equations in polynomial form and computing their resultant. This step is then followed by a second resultant constructed from the first. However, this approach often yields unwieldy expressions. In addition, one has to check the results for spurious solutions. These do not occur in the two-point-mass lens, but they do appear in all the triple-lens models studied further below. As an alternative, we present in § 3.3 and § 3.4 two efficient numerical methods for mapping the boundaries. These can be used in models with at least one scale-defining parameter, such as the component separation s in the two-point-mass lens. In the first method we find the roots of the first resultant condition, while in the second we utilize the Jacobian scaling properties described by Danˇek & Heyrovsky´ (2015). Both methods are free of spurious solutions. 3.2. Analytical boundaries computed by resultant method The method described here was pioneered in the context of critical-curve topology map- ping by Erdl & Schneider (1993) and Witt & Petters (1993). Multiplying the saddle-point equation (6) by the product of its denominators yields n n (cid:88) (cid:89) p (z) = µ (z −z )3 = 0, (8) sadd j k j=1 k=1,k(cid:54)=j a polynomial of degree 3n−3. In a similar manner we convert the critical-curve equation (2) to n n n (cid:89) (cid:88) (cid:89) p (z) = (z −z )2 −e2iφ µ (z −z )2 = 0, (9) crit k j k k=1 j=1 k=1,k(cid:54)=j a polynomial of degree 2n for any value of the parameter φ. The analytical condition for the existence of a common root of p (z) and p (z) is sadd crit Res (p ,p ) = 0, (10) z sadd crit – 9 – where the resultant Res (f,g) of two polynomials f, g is a function of their coefficients. It z may be computed by evaluating the determinant of the Sylvester or B´ezout matrices, as described in Appendix A. The expression obtained from equation (10) is a polynomial in terms of e2iφ. If we denote w = e2iφ, we can write the result as m (cid:88) p (w) = a wj = 0, (11) res j j=0 where the degree of the polynomial m ≤ 3n−3. The boundary condition is now equivalent to the condition for p to have a root on the unit circle. res In order to obtain the condition purely in terms of the lens parameters, Witt & Petters (1993) suggested the following approach for the two-point-mass lens. For a root along the unit circle wj = w−j, so that if we take the complex conjugate of equation (11) and multiply it by wm, we get another polynomial equation m (cid:88) p (w) = a¯ wj = 0. (12) conj m−j j=0 Along the boundary in parameter space, polynomials p and p must have a common res conj root, thus Res (p ,p ) = 0 (13) w res conj yields the sought condition in terms of lens parameters. We point out that equation (13) presents a single constraint in parameter space. Hence for two-parameter models (such as the two-point-mass lens or the triple-lens models described in § 5.2 – § 5.5) it generally describes a set of curves, for three-parameter models a set of surfaces, etc. Weillustratetheapproachhereonthecaseofthetwo-point-masslens(Erdl&Schneider 1993;Witt&Petters1993),parameterizedbythefractionalmassofonecomponentµ ∈ (0,1) and the separation between the components s > 0. If we align the lens with the real axis and place the components symmetrically about the origin, their positions and masses are {z ,z } = {−s/2,s/2}, and {µ ,µ } = {µ,1−µ}. The polynomial equations for p and 1 2 1 2 sadd p are crit 3 3 1 p (z) = z3 + s(1−2µ)z2 + s2z + s3(1−2µ) = 0 (14) sadd 2 4 8 and 1 1 p (z) = z4 − (s2 +e2iφ)z2 −s(1−2µ)e2iφz + s2(s2 −4e2iφ) = 0. (15) crit 2 16 Using them in equation (10) leads to (cid:2) (cid:3) Res (p ,p ) = −s6µ2(1−µ)2 e6iφ −3s2(1−9µ+9µ2)e4iφ +3s4e2iφ −s6 = 0. (16) z sadd crit – 10 – The factors in front of the square brackets are non-zero for any genuine two-point-mass lens. If we set e2iφ = w, the term in the brackets yields the polynomial equation p (w) = w3 −3s2(1−9µ+9µ2)w2 +3s4w−s6 = 0. (17) res Following equation (12) we construct p (w) = −s6w3 +3s4w2 −3s2(1−9µ+9µ2)w+1 = 0. (18) conj The resultant obtained from equation (13) can be factorized as follows: (cid:2) (cid:3)(cid:2) (cid:3) 1−3s2(1−9µ+9µ2)+3s4 −s6 1+3s2(1−9µ+9µ2)+3s4 +s6 ×(cid:2)1−3s4 +3s8(1−9µ+9µ2)−s12(cid:3)2 = 0. (19) At least one of these three square brackets thus has to be equal to zero. The first bracket in equation (19) is equal to p (1), with w = 1 corresponding to φ = 0. res Therefore, it must include any transitions occurring on the critical curve along the real axis, in this case the passage of the critical curve through the central saddle point. Taken as a polynomial in s, the first bracket has a single real positive root (Erdl & Schneider 1993) (cid:104)√ (cid:112) (cid:105)3/2 s = 3 µ+ 3 1−µ , (20) w which is the boundary between the wide and intermediate topologies. The second bracket is equal to −p (−1), with w = −1 corresponding to φ = π/2. res √ There is only a single zero point of this expression in parameter space, [µ, s] = [0.5, 0.5], whichcorrespondstotheintermediate–closesplittingalongtheimaginaryaxisofthecritical curve of an equal-mass lens. The third bracket corresponds to topology transitions that occur at any other values of φ on the critical curve. Here they describe the passage of the critical curve through the pair of saddle points off the real axis. Taken as a polynomial in s, the third bracket has a single real positive root (Erdl & Schneider 1993; Rhie & Bennett 1999) (cid:104)√ (cid:112) (cid:105)−3/4 s = 3 µ+ 3 1−µ = s−1/2, (21) c w which is the boundary between the intermediate and close topologies. This boundary in fact passes through the single zero point of the second bracket in equation (19) at µ = 0.5 as well. The two curves given by equations (20) and (21) thus fully describe the division of the two-point-mass lens parameter space according to critical-curve topology.